2,877 reputation
627
bio website combinatorialgames.wordpress.…
location United States
age 26
visits member for 2 years, 4 months
seen 2 days ago

I have an amateur interest in combinatorial game theory and rarely update a blog with some basic exposition on the subject (see website).

If you need to contact me, use the e-mail address at this link.


May
26
comment How exactly is $i=\sqrt{-1}$ related to $\mathbb{C}$ being a closed algebraic field?
It's important to note (and the links clarify this) that the Quaternions are not what you get when you have the same goal (algebraic closure) but happen to be working with matrices, but rather something reminiscent of the complex numbers you can get when you change/weaken the goal.
May
26
comment How exactly is $i=\sqrt{-1}$ related to $\mathbb{C}$ being a closed algebraic field?
@Nikos $\sqrt\pi$ is the number that you square to get $\pi$. Every positive number has a positive square root: If you believe it for rationals, just take the limit of a sequence of rational numbers whose square is not quite big enough, but whose squares tend to the number in question.
May
26
comment Are there non-zero combinatorial games of odd order?
The proof of "no (nonzero) games of odd order" is too long to fairly reproduce here. I suppose it's worth mentioning that the key result (which is not so easy to prove) is that if $G$ has finite order and birthday $n$, then $2^nG=0$.
Apr
19
comment Are there 3 trig functions or are there 6 trig functions?
This appears to assume cosine is positive.
Mar
26
comment Necessary/sufficient conditions for an infinite product to be exactly equal to $1$
@pbs it was a typo since the text said equal to zero.
Mar
25
comment Necessary/sufficient conditions for an infinite product to be exactly equal to $1$
@sabyasachi yes, but the sequence 3,1,5,1,7,1,... diverges due to oscillation.
Mar
25
comment Necessary/sufficient conditions for an infinite product to be exactly equal to $1$
@sabyasachi I don't think those terms have product limit 1, but Daniel ' s example is fine.
Mar
25
comment Necessary/sufficient conditions for an infinite product to be exactly equal to $1$
The limit of the terms better be 1, but if you're strictly monotonically increasing or decreasing, then all terms are on one side of 1, which means the product won't be 1.
Mar
21
comment $\max_{y} \min_{x} f(x,y)$ as motif for exploring mathematics
@alex.jordan for every fixed $ y $, the function of a single variable $ g_y (x)=f (x, y) $ may have a minimum value, but different $ y $ s will give different minima. You can collect them all up into a function of $ y$ called $\min_x f (x, y) $. Since the min of $x^2/2-\pi x $ is $-\pi^2/2$, and similarly if we replace $\pi $ by any arbitrary number $ y $, lilinjn's expression makes sense
Mar
13
comment Is there a term for parentheses and brackets in equations?
@andre would you be interested in posting that as an answer to remove this from the unanswered list?
Mar
13
comment Prove that a polynomial has at least one nonreal complex root
For this problem, the technique is to give Descartes' Rule of Signs a better chance, but there is also a generalization of the discriminant method. If your quintic is $x^5+px^3+qx^2+rx+s$ and the discriminant is positive, then it has one real root precisely if at least one of the following is nonpositive: $-p$, $40rp-12p^3-45q^2$, and $12p^4r-4p^3q^2-40p^2qs-88p^2r^2+117pq^2r+125ps^2-27q^4-300qrs+160r^3$. These formulas can be found in demonstrations.wolfram.com/…, which also cites "A Complete Discrimination System for Polynomials".
Mar
13
comment Prove that a polynomial has at least one nonreal complex root
@BohanLu depressing (making a linear substitution to eliminate the term of second-highest order) a polynomial is a key first step in solving polynomials. Depressing a quadratic is basically "completing the square", depressing a cubic/quartic is the first step to solving those equations, and even though quintics don't often have solutions in radicals, depressing them makes them simpler for solving/characterization. (Continued below:
Mar
12
comment Simple combinatorics question - caught off guard!
@Darrin I haven't read the book, but this is a direct proof as opposed to a proof by contradiction (or even a proof by induction).
Mar
12
comment 2nd Question in introductory probability
It would be much easier to point you in the right direction if you posted your thoughts on the problem, where this problem comes from, etc.
Mar
11
comment Player statistics as estimate of surreal number of game
@Skatche To your first comment: No; I would simply set up things like $G+99$ and $G+100$ and if Right often won the former and Left the latter, I might have grounds to bound the value of $G$ between $-99$ and $-100$. Second comment: I had intended the numbers to be in canonical form, so that right could have no moves at any subposition (for the most extreme case), but you bring up a good point: looking at surreal values won't distinguish between $\lbrace 99\mid101\rbrace$ and, say, $\lbrace 99,-1,-2,-3,-23897,-5897\mid101,100\rbrace$, so that the values don't account for chances to slip up.
Mar
11
comment Summation notation for time series
@Stuartzz I'm glad it was helpful. If you feel my answer was what you were looking for, you can accept it to let people know the question has been answered.
Mar
9
comment Winning a restricted game of Nim?
@zyx It must have period (a factor of 4) for the reason you state. In this particular case, the sequence is simply 0101... As an aside, since these are octal games, this can also be looked up. The new game audiFanatic proposes is .303, which is analogous to .05 (a game in which you can remove an entire pile of size 2, or remove two from a pile of size at least 4 to leave two separate piles) which has sequence 00101010...
Mar
9
comment Winning a restricted game of Nim?
As an aside, I would say that there are more accessible resources than Winning Ways for most of the material contained within it. For impartial games alone, there's Tom Ferguson's "Game Theory", and for an introductory book on CGT, Lessons in Play will be an easier read, I feel.
Mar
9
comment LaTeX/TeX Vs. Mathematica for Typesetting
Regarding Scientific Word, it has taken some years, but I would now say that LyX (which is free) is better.
Mar
9
comment Intuition about the Riemann and Ricci tensors
I couldn't quite find an applet that would do what I wanted, although things like torus.math.uiuc.edu/jms/java/dragsphere were close. Basically, I would recommend doing an explicit approximate calculation of the Riemann tensor (and then sectional curvature) for a sphere with the two directions starting out orthogonal. Only from working through the definitions with a toy case like that did I get the sense that "positive Gaussian curvature really should have positive sectional curvature as defined in terms of the Riemann tensor", which really helped me feel better about it.